PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 3, March 1997, Pages 629–634 S 0002-9939(97)03581-8 GROUP ALGEBRAS WHOSE UNITS SATISFY A GROUP IDENTITY ANTONIO GIAMBRUNO, SUDARSHAN SEHGAL, AND ANGELA VALENTI (Communicated by Ronald M. Solomon) Abstract. Let FG be the group algebra of a torsion group over an infinite field F . Let U be the group of units of FG. We prove that if U satisfies a group identity, then FG satisfies a polynomial identity. This confirms a conjecture of Brian Hartley. 1. Introduction The unit group U = U (FG) is said to satisfy a group identity if there exists a nontrivial word w(x 1 ,...,x m ), in the free group generated by x 1 ,...,x m , such that w(u 1 ,...,u m ) = 1 for all u i ∈ U . Hartley suggested the following Conjecture 1.1. If G is a torsion group and U (FG) satisfies a group identity, then FG satisfies a polynomial identity. We recall that FG is said to satisfy a polynomial identity (PI ) if there exists a nonzero polynomial f (y 1 ,...,y n ) ∈ F {y 1 ,...,y n }, in noncommuting variables such that f (α 1 ,...,α n ) = 0 for all α i ∈ FG. Group algebras satisfying a PI were classified by Passman and Isaacs-Passman. This conjecture was first studied by Warhurst [10] who investigated special words satisfied by U (FG). Pere Menal [5] suggested a possible solution for some p-groups. We are able to use his construction. Giambruno-Jespers-Valenti [4] settled the case where G has no p-element if p is the characteristic of F . Further Goncalves- Mandel [3] and Dokuchaev-Goncalves [1] studied group algebras when U satisfies a semigroup identity. There have been many papers classifying groups so that U satisfied a special group identity like (u 1 ,...,u n ) or (u n ,v) and many others. The main result of this paper is the following theorem. Theorem. Suppose that F is an infinite field and that G is a torsion group. If U (FG) satisfies a group identity, then FG satisfies a polynomial identity. 2. Some lemmas We write (u, v)= u −1 v −1 uv for the multiplicative commutator and [x, y]= xy − yx for the additive commutator. We denote by Δ(G, N ) the kernel of FG → Received by the editors June 26, 1995. 1991 Mathematics Subject Classification. Primary 16S34; Secondary 20C05. Research supported by NR and MURST of Italy and NSERC of Canada. c 1997 American Mathematical Society 629 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use